Question Bank • QB_EM_DE_011 MCQ • 1 Mark

Section 1: Engineering Mathematics

Differential Equations

Subtopic: Laplace transforms and their inverse

Question Statement:

The Laplace transform of the delayed unit step function $u(t - a)$ for $a > 0$ is:

Options:

  • (A) $\frac{e^{-as}}{s}$
  • (B) $\frac{e^{as}}{s}$
  • (C) $\frac{1}{s - a}$
  • (D) $\frac{e^{-as}}{s^2}$

Answer Key:

A

Step-by-Step Solution & Explanation

By the definition of the Laplace transform: $\mathcal{L}\{u(t - a)\} = \int_0^\infty e^{-st} u(t - a) \, dt = \int_a^\infty e^{-st} \, dt = \left[\frac{e^{-st}}{-s}\right]_a^\infty = \frac{e^{-as}}{s}$ Thus, option A is correct.
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